Strategic Insights: When to Play a Knight in Settlers of Catan
To excel at Settlers of Catan, it is crucial to understand the expected value of common actions within the game. Understanding expected value is particularly useful - and often overlooked - in deciding when to play a knight.
Understanding Expected Value
Expected value is the average result of an action repeated multiple times. In Catan, it is easy to compare the expected value of placing a settlement on different hexes by comparing the total number of pips. For example, a hex with 5 pips (like a 6-hex spot) will, on average, produce five times more resources than a 1-pip hex (like a 2-hex spot).
Most players use this pip system to help them decide where to place their initial settlements, but not much else. However, learning to apply the pip system to other decisions within the game can give you a significant edge. For example, you can also use pips to determine when to play a knight.
The Knight Dilemma: Before or After the Dice Roll?
A common strategic decision in Catan is choosing when to play the knight - before or after rolling the dice. Conventional strategy advises waiting until after the roll if you have seven cards. The reasoning is straightforward: playing the knight before rolling will increase your card count above seven, risking a penalty if a seven is rolled. But this isn't always the best strategy.
There are situations where it's beneficial to play a knight even if it increases your card count beyond seven. By applying some basic expected value calculations, I've formulated a practical guideline to help you decide the right time to play your knight.
The Expected Value of Playing a Knight
Playing a knight with seven cards impacts your game in three ways:
Takes you above seven cards: Increasing the chance of losing cards by rolling a seven. There's a 16.67% chance of rolling a seven, leading to a potential loss of four cards. On average, this translates to losing 0.67 cards (0.17 probability times 4 cards).
Unblocks your hex: Increasing the chance of gaining resources if your hex’s number is rolled. The expected value of this varies based on your hex's production. For instance, unblocking a 6-hex with a city can yield an expected 0.28 cards (0.14 probability times 2 for a city).
Blocks opponent's hex: decreasing the chance your opponents will gain resources. As with unblocking, the expected value of this depends on the production of the hex being blocked. Blocking a 5-hex with two cities could, on average, prevent opponents from gaining 0.44 cards (0.11 probability times 4 for 2 cities).
These factors combine to determine the overall expected value of playing a knight:
-0.67 + 0.28 + 0.44 = 0.05
In other words, you can expect to gain 0.05 cards relative to your opponents by playing your knight before rolling in this scenario. The table below details the expected card gain or loss for different scenarios, with red cells indicating a net loss and green cells a net gain.
Expected value of playing a knight 👉 View Google Sheet
A Simple Rule of Thumb
Observing the table above, a simple rule of thumb emerges: in most instances, you should play the knight before rolling when you can block 20 or more pips of production.
To provide a few practical examples for when you have seven cards:
Play a knight to block 4X production on a 6 or 8-hex.
Play a knight to block 5X production on a 5 or 9-hex.
Generally, avoid using a knight to block 4 or 10-hex.
Applying These Guidelines in Gameplay
While these guidelines are based on a simplified model (assuming equal value for all resources and no shared hexes with opponents), they provide a useful framework for decision-making. If you're considering moving the knight to block less than 20 pips of production, ensure you have a strong strategic reason for doing so.
